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Mirrors > Home > ILE Home > Th. List > foco2 | Unicode version |
Description: If a composition of two functions is surjective, then the function on the left is surjective. (Contributed by Jeff Madsen, 16-Jun-2011.) |
Ref | Expression |
---|---|
foco2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 981 | . 2 | |
2 | foelrn 5654 | . . . . . 6 | |
3 | ffvelrn 5553 | . . . . . . . . . 10 | |
4 | 3 | adantll 467 | . . . . . . . . 9 |
5 | fvco3 5492 | . . . . . . . . . 10 | |
6 | 5 | adantll 467 | . . . . . . . . 9 |
7 | fveq2 5421 | . . . . . . . . . . 11 | |
8 | 7 | eqeq2d 2151 | . . . . . . . . . 10 |
9 | 8 | rspcev 2789 | . . . . . . . . 9 |
10 | 4, 6, 9 | syl2anc 408 | . . . . . . . 8 |
11 | eqeq1 2146 | . . . . . . . . 9 | |
12 | 11 | rexbidv 2438 | . . . . . . . 8 |
13 | 10, 12 | syl5ibrcom 156 | . . . . . . 7 |
14 | 13 | rexlimdva 2549 | . . . . . 6 |
15 | 2, 14 | syl5 32 | . . . . 5 |
16 | 15 | impl 377 | . . . 4 |
17 | 16 | ralrimiva 2505 | . . 3 |
18 | 17 | 3impa 1176 | . 2 |
19 | dffo3 5567 | . 2 | |
20 | 1, 18, 19 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 wral 2416 wrex 2417 ccom 4543 wf 5119 wfo 5121 cfv 5123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fo 5129 df-fv 5131 |
This theorem is referenced by: (None) |
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